Abstract
Benjamini and Papasoglou (2011) showed that planar graphs with uniform polynomial volume growth admit 1-dimensional annular separators: The vertices at graph distance R from any vertex can be separated from those at distance 2R by removing at most O(R) vertices. They asked whether geometric d-dimensional graphs with uniform polynomial volume growth similarly admit \((d-1)\)-dimensional annular separators when \(d>2\). We show that this fails in a strong sense: For any \(d\geqslant 3\) and every \(s\geqslant 1\), there is a collection of interior-disjoint spheres in \(\mathbb {R}^d\) whose tangency graph G has uniform polynomial growth, but such that all annular separators in G have cardinality at least \(R^s\).
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Notes
Depiction of a random triangulation is due to Nicolas Curien.
References
Angel, O.: Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13(5), 935–974 (2003)
Bonk, M., Kleiner, B.: Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math. 150(1), 127–183 (2002)
Benjamini, I., Papasoglu, P.: Growth and isoperimetric profile of planar graphs. Proc. Am. Math. Soc. 139(11), 4105–4111 (2011)
Benjamini, I., Schramm, O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6, \(\#\,23\) (2001)
Ding, J., Gwynne, E.: The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds. Commun. Math. Phys. 374(3), 1877–1934 (2020)
Ebrahimnejad, F., Lee, J.R.: On planar graphs of uniform polynomial growth. Probab. Theory Relat. Fields 180(3–4), 955–984 (2021)
Gwynne, E., Holden, N., Sun, X.: A mating-of-trees approach for graph distances in random planar maps. Probab. Theory Relat. Fields 177(3–4), 1043–1102 (2020)
Krikun, M.A.: Uniform infinite planar triangulation and related time-reversed critical branching process. J. Math. Sci. 131(2), 5520–5537 (2005)
Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36(2), 177–189 (1979)
Miller, G.L., Teng, Sh.-H., Thurston, W., Vavasis, S.A.: Separators for sphere-packings and nearest neighbor graphs. J. ACM 44(1), 1–29 (1997)
Miller, G.L., Teng, Sh.-H., Thurston, W., Vavasis, S.A.: Geometric separators for finite-element meshes. SIAM J. Sci. Comput. 19(2), 364–386 (1998)
Plotkin, S., Rao, S., Smith, W.D.: Shallow excluded minors and improved graph decompositions. In: 5th Annual ACM-SIAM Symposium on Discrete Algorithms (Arlington 1994), pp. 462–470. ACM, New York (1994)
Spielman, D.A., Teng, Sh.-H.: Spectral partitioning works: planar graphs and finite element meshes. Linear Algebra Appl. 421(2–3), 284–305 (2007)
Teng, Sh.-H.: Combinatorial aspects of geometric graphs. Comput. Geom. 9(4), 277–287 (1998)
Acknowledgements
This research was partially supported by NSF CCF-2007079 and a Simons Investigator Award.
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Ebrahimnejad, F., Lee, J.R. Non-Existence of Annular Separators in Geometric Graphs. Discrete Comput Geom 71, 627–645 (2024). https://doi.org/10.1007/s00454-023-00519-8
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DOI: https://doi.org/10.1007/s00454-023-00519-8